On the power of white pebbles

We construct a family (G _p |p) of directed acyclic graphs such that any black pebble strategy forG _p requiresp ² pebbles whereas 3p+1 pebbles are sufficient when white pebbles are allowed.Supported by the National Science Foundation under contract no. DCR-8407256 and by the office of Naval Research under contract no. N0014-80-0517.     

Expected length of the longest common subsequence for large alphabets

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that E[L]/n converges to a constant γ_k. We prove a conjecture of Sankoff and Mainville from the early 1980s claiming that as k→∞.     

The asymmetric leader election algorithm: Number of survivors near the end of the game

The following classical asymmetric leader election algorithm has obtained quite a bit of attention lately. Starting with n players, each one throws a coin, and the k of them which have each thrown a head (with probability q) go on, and the leader will be found amongst them, using the same strategy. Should nobody advance, the party will repeat the procedure. One of the most interesting parameter here is the number J (n) of rounds until a leader has been identified. In this paper we investigate, in the classical leader election algorithm, what happens near the end of the game, namely we fix an integer κ and we study the behaviour of the number of survivors L at level J (n) ? κ. In our asymptotic analysis (for n → ∞) we are focusing on the limiting distribution functions. We also investigate what happens, if the parameter p = 1 ? q gets small (p → 0) or large (p → 1). We use three e?cient tools: an urn model, a Mellin-Laplace technique for harmonic sums and some asymptotic distributions related to one of the extreme-value distributions: the Gumbel law. This study was motivated by a recent paper by Kalpathy, Mahmoud and Rosenkrantz, where they consider the number of survivors S_n,t, after t election rounds, in a broad class of fair leader election algorithms starting with n candidates.     

The complexity of detecting crossingfree configurations in the plane

The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.Klaus Jansen acknowledges support by the Deutsche Forschungsgemeinschaft. Gerhard J. Woeginger acknowledges support by the Christian Doppler Laboratorium für Diskrete Optimierung.     

The complexity of computing the tutte polynomial on transversal matroids

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.     

Sensitivity vs. block sensitivity of Boolean functions

Senstivity and block sensitivity are important measures of complexity of Boolean functions. In this note we exhibit a Boolean function ofn variables that has sensitivity and block sensitivity (n). This demonstrates a quadratic separation of the two measures.c/o László Babai     

On the recognition complexity of some graph properties

By applying a topological approach due to Kahn, Saks and Sturtevant, we prove that all decreasing graph properties consisting of bipartite graphs only are elusive. This is an analogue to a well-known result of Yao.     

Recursive constructions of secure codes and hash families using difference function families

To protect copyrighted digital data against piracy, codes with different secure properties such as frameproof codes, secure frameproof codes, codes with identifiable parent property (IPP codes), traceability codes (TA codes) are introduced. In this paper, we study these codes together with related combinatorial objects called separating and perfect hash families. We introduce for the first time the notion of difference function families and use these difference function families to give generalized recursive techniques that can be used for any kind of secure codes and hash families. We show that some previous recursive techniques are special cases of these new techniques.     

Spectral properties of threshold functions

We examine the spectra of boolean functions obtained as the sign of a real polynomial of degreed. A tight lower bound on various norms of the lowerd levels of the function's Fourier transform is established. The result is applied to derive best possible lower bounds on the influences of variables on linear threshold functions. Some conjectures are posed concerning upper and lower bounds on influences of variables in higher order threshold functions.Supported by an Eshkol fellowship, administered by the National Council for Research and Development—Israel Ministry of Science and Development.     

On rank vs. communication complexity

This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.A preliminary version of this paper appeared in [10].This work was supported by USA-Israel BSF grant 92-00043 and by a Wolfeson research award administered by the Israeli Academy of Sciences.This work was supported by USA-Israel BSF grant 92-00106 and by a Wolfeson research award administered by the Israeli Academy of Sciences.     

On computing majority by comparisons

The elements of a finite setX (of odd cardinalityn) are divided into two (as yet unknown) classes and a member of the larger class is to be identified. The basic operation is to test whether two objects are in the same class. We show thatn-B(n) comparisons are necessary and sufficient in worst case, whereB(n) is the number of 1's in the binary expansion ofn.Supported in part by NSF grant DMS87 03541 and Air Force Office of Scientific Research grant AFOSR-0271.     

The elimination procedure for the competition number is not optimal

Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V(D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs (x,z) and (y,z) are both present in D. The competition number k(G) for an undirected graph G is the least number r such that there exists an acyclic digraph F on |V(G)|+r vertices where C(F) is G along with r isolated vertices. Kim and Roberts [The Elimination Procedure for the Competition Number, Ars Combin. 50 (1998) 97-113] introduced an elimination procedure for the competition number, and asked whether the procedure calculated the competition number for all graphs. We answer this question in the negative by demonstrating a graph where the elimination procedure does not calculate the competition number. This graph also provides a negative answer to a similar question about the related elimination procedure for the phylogeny number introduced by the current author in [S.G. Hartke, The Elimination Procedure for the Phylogeny Number, Ars Combin. 75 (2005) 297-311].     

Subwords in Reverse-Complement Order

We examine finite words over an alphabet of pairs of letters, where each word w₁w₂ ... w_t is identified with its reverse complement where ( ). We seek the smallest k such that every word of length n, composed from Γ, is uniquely determined by the set of its subwords of length up to k. Our almost sharp result (k~ 2n = 3) is an analogue of a classical result for “normal” words. This problem has its roots in bioinformatics. Received October 19, 2005     

Abelian repetitions in partial words

We study abelian repetitions in partial words, or sequences that may contain some unknown positions or holes. First, we look at the avoidance of abelian pth powers in infinite partial words, where p>2, extending recent results regarding the case where p=2. We investigate, for a given p, the smallest alphabet size needed to construct an infinite partial word with finitely or infinitely many holes that avoids abelian pth powers. We construct in particular an infinite binary partial word with infinitely many holes that avoids 6th powers. Then we show, in a number of cases, that the number of abelian p-free partial words of length n with h holes over a given alphabet grows exponentially as n increases. Finally, we prove that we cannot avoid abelian pth powers under arbitrary insertion of holes in an infinite word.     

Bit strings withoutq-separation

LetB(n, q) denote the number of bit strings of lengthn withoutq-separation. In a bit string withoutq-separation no two 1's are separated by exactlyq – 1 bits.B(n, q) is known to be expressible in terms of a product of powers of Fibonacci numbers. Two new and independent proofs are given. The first proof is by combinatorial enumeration, while the second proof is inductive and expressesB(n, q) in terms of a recurrence relation.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

The asymptotic number of graphs not containing a fixed color-critical subgraph

For a finite graphG letForb(H) denote the class of all finite graphs which do not containH as a (weak) subgraph. In this paper we characterize the class of those graphsH which have the property that almost all graphs inForb(H) are -colorable. We show that this class corresponds exactly to the class of graphs whose extremal graph is the Turán-graphT _n ().An earlier result of Simonovits (Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions,Discrete Math. 7 (1974), 349–376) shows that these are exactly the (+1)-chromatic graphs which contain a color-critical edge.     

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.